differentiation rules

Here follow the rules for differentiating common functions of a single real variable that every student of calculus should know. In each case the variable with respect to which the derivative is being taken is \(x\), both \(u\) and \(v\) are functions of \(x\),and \(c\) is a constant. Note that \(u^{\prime}\) should be understood as \(\displaystyle\frac{du}{dx}\).

\[ \begin{eqnarray*} \frac{d}{dx}\left( c \right) & = & 0 \\ & & \\ \frac{d}{dx}\left( x \right) & = & 1 \\ & & \\ \frac{d}{dx}\left( cu \right) & = & cu^{\prime} \\ & & \\ \frac{d}{dx}\left( u \pm v \right) & = & u^{\prime} + v^{\prime} \\ & & \\ \frac{d}{dx}\left( uv \right) & = & u^{\prime}v + uv^{\prime} \\ & & \\ \frac{d}{dx}\left( \frac{u}{v} \right) & = & \frac{u^{\prime}v - uv^{\prime}}{v^2} \\ & & \\ \frac{d}{dx}\left( u^n \right) & = & nu^{n-1}u^{\prime} \\ & & \\ \frac{d}{dx}\left( \ln u \right) & = & \frac{u^{\prime}}{u} \\ & & \\ \frac{d}{dx}\left( \sin u \right) & = & (\sin u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \cos u \right) & = & (-\sin u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \tan u \right) & = & \left(\sec^2 u \right) u^{\prime} \\ & & \\ \frac{d}{dx}\left( \cot u \right) & = & -\left( \csc^2 u\right) u^{\prime} \\ & & \\ \frac{d}{dx}\left( \sec u \right) & = & (\sec u \tan u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \csc u \right) & = & -(\csc u \cot u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \arcsin u \right) & = & \frac{u^{\prime}}{\sqrt{1-u^2}} \\ & & \\ \frac{d}{dx}\left( \arccos u \right) & = & \frac{-u^{\prime}}{\sqrt{1-u^2}} \\ & & \\ \frac{d}{dx}\left( \arctan u \right) & = & \frac{u^{\prime}}{1+u^2} \\ & & \\ \frac{d}{dx}\left( \mbox{arccot } u \right) & = & \frac{-u^{\prime}}{1+u^2} \\ & & \\ \frac{d}{dx}\left( \mbox{arcsec } u \right) & = & \frac{u^{\prime}}{|u|\sqrt{u^2-1}} \\ & & \\ \frac{d}{dx}\left( \mbox{arccsc } u \right) & = & \frac{-u^{\prime}}{|u|\sqrt{u^2-1}} \\ & & \\ \end{eqnarray*} \]